Monday, February 13, 2006

It's good to finally have something up we can use. You have some very interesting questions there about the way we "describe" dimensions. I use the word describe because I believe that we do not build dimensions per say, but rather, we build descriptions of the dimensions. The problem I believe is that we mix description with described object. Dimensions allow us to be able to have something we can imagine about space [not the 3 D space]:)

The real question which you raise is, can we have a different description that works as well or maybe better? Another way of asking the same question is: what is wrong with the current description? Does it for example allow us to clearly define infinities? I think not and this ties back to our last conversation about the set of "all" infinities. What is it? how is it constructed? Does such a set even exist?

PS: Adding to your list of some of the things we establish last time, we "proved" that there are as many rational numbers as there are irrational numbers by creating injections both ways.

We did not quite reach a conclusion on the number of infinties as we tried to define what the set of infinities could look like. One of the ideas was that it is itself an infinite set...puzzling...


SLL

Monday, February 06, 2006

I'm here - it's taken too long, but now is now :) Anyway, something struck me today while I was driving... however profound or mundane it is, and that was, is our present construction/conception of physical dimension necessarily correct. I mean, we start with a 0 dimensional point, build a 1 dimentional line by "lining up" points (although not necessarily in a line... consider a circle), then build 2 dimensional surfaces by "swiping" those "lines" (in quotes because they are not necessarily lines as noted), then 3 dimensional objects by "pushing" surfaces... what bothered me was that we seem to build fundamentally from those constructions (which are admittedly ok for me intuitively)... the question that came was... can we start in a different way and construct "dimensions" which are fundamentally different from those we now use intuitively? I have a feeling this is either something subtle and interesting, or simply nothing to consider at all... perhaps the current construction encapsulates all of what even I am thinking of as dimensionality... but something about the limitation of starting with points and getting to "lines" and building from there seems to be something of a choice... are there other creative choices for "starting" which yield parallels to what we now think of as dimensions? I think either there are OR the concept of dimension is something more fundamental as an idea, than simply to serve as an important construction for physics.

PS - we still have much more to discuss regarding the number of infinities which exist :) At our last meeting, we discussed the following:
1. things that are the size of the naturals (0, 1, 2, 3, 4, ...) - also same size as integers.
2. things that are the size of the reals (which is like the power set - set of all subsets - of the naturals)

We discussed much more, and I want to find the link to the formalism on that matter we found last time... there was a conjecture there that there were only "so many" infinities if I remember correctly.

Talk soon,

JLS